My question originates from the book of Silverman "The Aritmetic of Elliptic Curves", 2nd edition (call it [S]). On p. 273 of [S] the author is considering an elliptic curve $E/K$ defined over a number field $K$ and he introduces the notion of a $v$-adic distance from $P$ to $Q$. This is done as follows:

Firstly, let's fix an absolute value (archimedean or not) $v$ of $K$ and a point $Q\in E(K_v)$ (here $K_v$ is the completion of $K$ at $v$). Next let's pick a function $t_Q \in K_v(E)$ defined over $K_v$ which vanishes at $Q$ to the order $e$ but has no other zeroes. Now the $v$-adic distance from $P \in E(K_v)$ to $Q$ is defined to be $d_v(P, Q) := \min (|t_Q(P)|_v^{1/e}, 1)$. We will say that $P$ goes to $Q$, written $P~\xrightarrow{v}~ Q$, if $d_v(P, Q) \rightarrow 0$. Later in the text (among other places in the proof of IX.2.2) the author considers a function $\phi\in K_v(E)$ which is regular at $Q$ and claims that this means that $|\phi(P)|_v$ is bounded away from $0$ and $\infty$ if $P~\xrightarrow{v}~ Q$.

I have a couple of questions about this:

How does one choose a $t_Q$ that works? In the footnote in [S] it is demonstrated how one could use Riemann-Roch to pick a $t_Q$ that has a zero only at $Q$. It seems to me however that such a procedure will not make sure that $t_Q$ is defined over $K_v$ since $K_v$ is not algebraically closed.

For $\phi$ as above which does not vanish nor has a pole at $Q$, how does one see that $|\phi(P)|_v$ is bounded away from $0$ and $\infty$ as $P~\xrightarrow{v}~ Q$?

Do these $d_v$ have anything to do with defining a topology on $E(K_v)$? I assume not, since I don't see how to make sense of it; but then on the other hand they are called "distance functions"...

2 Answers
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You can choose $t_Q$ to be defined over $K_v$, since the divisor $n(Q_v)$ is defined over $K_v$, and for large enough $n$ there will be a global section. Note that Riemann-Roch works over non-algebraically closed fields this way. Or you can choose a basis defined over some finite Galois extension of $K_v$, and then taking appropriate linear combinations of the Galois conjugates, get a function defined over $K_v$. See Proposition II.5.8 in [S].

The function defined in the text is only a reasonable "distance function" in the sense that it measures the distance from $P$ to the fixed point $Q$. For the purposes of this proof, that's fine. If you want to define the $v$-adic topology, you need to be a little more careful. Locally around $Q$ you could use $$d_v(P_1,P_2)=min(|t_Q(P_1)-t_Q(P_2)|^{1/e},1)$$, but that still only works in a neighborhood of $Q$, i.e., in a set $$\{P : d_v(P,Q)<\epsilon\}$$ for a sufficiently small $\epsilon$. Using local height functions, more precisely the local height relative to the diagonal in $E(K_v)\times E(K_v)$, one gets a "good" distance function that is defined everywhere. See for example Lang's Fundamentals of Diophantine Geometry or the book Diophantine Geometry that Hindry and I wrote for the general construction of local height functions.

Some complement to Joe Silverman's answer. Any algebraic variety over $K_v$ (or any topological field) has a canonical topology induced by that of the base field. This topology can be defined by a distance (far from to be unique). Over $\mathbb{P}^n_{K_v}$, a distance can be given (once a system de coordinates is fixed) by
$$ d((x_0,\dots, x_n), \ (y_0, \dots, x_n))= \dfrac{\max_{i, j} \lbrace |x_iy_j-x_jy_i|_v \rbrace}{(\max_i\lbrace |x_i|_v\rbrace\max_j\lbrace |y_j|_v \rbrace)}
$$
This is a non-archimedean distance. Concretely, one can see that if there exists an index $r$ such that $|x_i|_v\le |x_r|_v$ and $|y_j|_v\le |y_r|_v$ for all $i, j$, then
$$d(x,y)=\max_i \lbrace |x_i/x_r - y_i/y_r|_v \rbrace. $$
Otherwise $d(x,y)=1$.

One can describe this distance as following: there is a canonical reduction map $\pi: \mathbb P^n(K_v) \to \mathbb P^n(\mathbb k_v)$ where $k_v$ is the residue field of $K_v$. This map consists, after dividing by a coordinate of maximal absolute value, in reducing the coordinates mod $m_v$.
The fiber $\pi^{-1}(P)$ of a rational point $P\in \mathbb P^n(\mathbb k_v)$ is just an open unit polydisk. Now if $\pi(x)=\pi(y)$, then $d(x,y)$ is the usual distance in the unit polydisk (maximum of the $|x_i-y_i|_v$), and $d(x,y)=1$ otherwise.

For any quasi-projective variety $X$ over $K_v$, the embedding in some projective space induces a distance on $X$ with the above distance on projective spaces.

If $X$ has a smooth quasi-projective model $\mathcal X$ such that canonical map $\mathcal X(O_v)\to X(K_v)$ is surjective (hence bijective), one can define a distance using the reduction map $\pi: X(K_v)\simeq \mathcal X(O_v)\to \mathcal X(k_v)$ similarly to the projective space (the fibers of $\pi$ are analytically isomorphic to a polydisk). If we embedd $\mathcal X$ into a projective space $\mathbb P^n_{O_v}$, then this distance is induced by the above distance on $\mathbb P^n$.

This applies to abelian varieties with their Néron models and the distance is canonical (compatible with homomorphisms of abelian varieties). I don't have access to the books of Lang and of Hindry-Silverman at home, I guess the distance described here has something to do with the good distance function that Joe alludes to.

Nice explanation. For archimedean absolute values, people often use instead the analogue of the classical chordal metric, which is $$d((x_0,\dots, x_n), \ (y_0, \dots, x_n))= \dfrac{\max_{i, j} \lbrace |x_iy_j-x_jy_i|_v \rbrace}{\sqrt{\sum |x_i|_v}\sqrt{\sum|y_j|_v}},$$ but the topology is the same, of course. However, the distance function is a local height relative to a divisor only on 1-dimensional varieties. In general, one can define local heights relative to closed subschemes, then one gets a distance function equivalent to the ones that Qing Liu has described.
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Joe SilvermanJul 30 '11 at 12:20

Typo, sorry, the denominator in that chordal metric formula should be $$\sqrt{\sum|x_i|_v^2}\sqrt{\sum|y_j|_v^2}.$$ I left off the squares.
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Joe SilvermanJul 30 '11 at 13:19